In general, working on the whole \(C^{\infty }(M,N)\) , it is very difficult to carry out a complete study of the second variation of a given biharmonic map. Therefore, in order to obtain some partial, but geometrically interesting results, it seems of interest to develop a method of investigation reducing in a suitable way the domain of the bienergy functional. More precisely, when a compact Lie group of isometries \(G\)  acts on both \(M\)  and \(N\) , we can restrict to the set \(\Sigma \)  of all symmetric points with respect to the natural action of \(G\)  on \(C^{\infty }(M,N)\) .
In this spirit, in the final section, because the Clifford torus \(\Phi (\mbox{${\mathbb {T}}$})\)  in \(\mathbb {S}^4\)  is a \(G\) -invariant submanifold with \(G={\rm SO}(2) \times {\rm SO}(2)\) , we compare our general results with those which can be deduced from the study of the second variation restricted to the set \(\Sigma \)  of all symmetric points of \(C^{\infty }(\mbox{${\mathbb {T}}$},\mathbb {S}^4)\)  with respect to the action of \(G\)  (see [1]}).
